作者linpotso (林柏佐)
看板Math
標題[分析] 泛函分析Banach Space
時間Thu May 5 22:40:26 2011
Peter Lax- Function Analysis
Chap. 15 Exercise 12. (接在closed graph theorem之後)
Show that for every infinite-dimensional Banach space there are
linear subspaces of finite codimension that are not closed.
(Hint: Use Zorn's Lemma)
請板上強者解答
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1F：推 ppia :fact: for each non-zero linear functional f, 05/06 14:24
2F：→ ppia :Ker f is a codimension 1 subspace. 05/06 14:25
3F：→ ppia :Lemma: if f is not bdd., then either is Ker f. 05/06 14:25
4F：→ ppia :(in fact, Ker(f) is dense in this space) 05/06 14:26
5F：→ ppia :"neither" 05/06 14:26
6F：→ ppia :therefore, it suffices to construct an unbdd f. 05/06 14:26
7F：→ ppia :this is always possible, since 05/06 14:27
8F：→ ppia :fact: a linear functional f can be constructed 05/06 14:27
9F：→ ppia :by specifying its values on a given basis. 05/06 14:28
10F：→ ppia :(NB: to prove the existence of linear space basis 05/06 14:28
11F：→ ppia :we need transfinite induction like Zorn's lemma 05/06 14:29
12F：→ linpotso :Thanks a lot, and I have some idea. 05/07 10:23
13F：→ linpotso :but how could i prove the space is not closed? 05/07 18:43
14F：→ linpotso :sorry, i can prove it, thank you 05/07 18:51
15F：推 keroro321 :推 05/07 22:19
16F：推 ppia :you can try to prove that Ker f is actually dense 05/08 12:32